Bristol remains one of the more delightfully inconvenient corners of the UK to reach in a lie-flat seat, and there is no nonstop option worth waiting for—which is precisely why the smart move for business class from Minneapolis to Bristol runs through Amsterdam, London Heathrow, or occasionally a Delta-Air France connection via Paris. Delta's transatlantic joint venture with Air France-KLM makes the Amsterdam routing especially seamless, with Delta One suites on the MSP-AMS long haul before a short KLM hop down to Bristol. Alternatively, British Airways offers its Club Suite (fitted on many 777 and 787 aircraft) out of London on the transatlantic leg, though travelers will need to connect through Heathrow's Terminal 5 and take a separate onward flight or train to Bristol, since BA doesn't operate mainline metal into BRS itself. American Airlines, tied to BA through the Oneworld joint business, gives you a third viable path, often routing through Heathrow or occasionally Philadelphia depending on seasonal scheduling.
Having flown this corridor more times than I can count, my strong preference is the Delta-KLM combination: Delta One's suite with a closing door on the MSP-AMS sector makes the eight-plus-hour transatlantic haul genuinely restorative, and Schiphol's compact layout makes tight connections far less stressful than Heathrow's sprawl. One insider tip that rarely surfaces in route guides: if your Heathrow connection has any slack, use the Elizabeth line rather than a connecting flight to reach Bristol—it's often faster door-to-door once you factor in security and boarding buffers at T5, and it sidesteps the frequent short-haul delays that plague BA's regional BRS rotations.
Seasonality here is pronounced. January through March and again in October and November consistently produce the softest fares and the best award availability, as Bristol's business travel demand dips outside the summer conference season and academic terms. Given how thin premium inventory can get on the connecting legs, especially through Amsterdam, setting up automated fare alerts is the only reliable way to catch a favorable business class fare before it disappears.